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Substituting(7.4)into(7.2)andgroupingterms
tQ i
A i θ i + 1 , j + 1 +
B i θ i , j + 1 +
C i θ i 1 , j + 1 = θ i , j 1 +
2
(7.5)
where
tK θ i + 1
tK θ i
A i =
2
C i =
2
B i =
1
A i
C i
z i
zz i
z i
zz i 1
Fromhereon,thesecondindexreferringtothenew
(
j
+
1
)
timestepwillbedropped
from the notation, with only the previous
(
j
1
)
time step indicated explicitly. To
usetheimplicit solutiontechnique,let
θ i =
D i +
E i θ i + 1
(7.6)
sothat
tQ i
A i θ i + 1 +
B i θ i +
C i (
D i 1 +
E i 1 θ i )= θ i , j 1 +
2
fromwhich
tQ θ i
=
A i θ i + 1 + θ i , j 1 +
2
C i D i 1
θ
(7.7)
i
B i +
C i E i 1
hencetherecursionrelationis
A i
E i =
(7.8)
A i
1
+
C i
(
1
E i 1
)
C i D i 1 θ i , j 1 +
tQ i
2
D i =
A i
1
+
C i (
1
E i 1 )
This approach holds as well for the momentumconservationequation in a rotating
reference frame provided the Coriolis term is treated as a source (time centered in
theleapfrogscheme).Usingcomplexnotationforthehorizontalvelocity(relativeto
undisturbedgeostrophicflow) in a horizontallyhomogeneousfluid,themomentum
balanceis
u t = τ z
if u
(7.9)
where
τ z is the verticalgradientof the kinematicReynoldsstress in the fluid. Note
that an additional source term (e.g., a constant or depth varying [“thermal wind”]
geostrophiccurrent)maybeaddedto theCoriolissourceterm.
7.2 Boundary Conditions
The starting point for the recursion relations (7.8) is provided by consideration of
theboundaryconditionsforthedifferenceformoftheconservationequationforthe
genericproperty
θ
.
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